Suppose $h_{i\overline{j}}$, where $1\leq i, j\leq n$, are real-analytic functions defined on $\mathbb{C}^n$ such that $H=(h_{i\overline{j}})$ is a Hermitian positive definite matrix. I wonder if it is always possible to find holomorphic functions $p_1,..., p_n$ defined on $\mathbb{C}^n$ such that $$\frac{\partial p_i}{\partial z_j}+\overline{\frac{\partial p_j}{\partial z_i}}=h_{i\overline{j}}.$$
The assumption that $H=(h_{i\overline{j}})$ is Hermitian is necessary, since $$\frac{\partial p_i}{\partial z_j}+\overline{\frac{\partial p_j}{\partial z_i}}=h_{i\overline{j}} =\overline{h_{j\overline{i}}} =\overline{\frac{\partial p_j}{\partial z_i}+\overline{\frac{\partial p_i}{\partial z_j}}}.$$ Also, I think it is always solvable when $n=1$, since it is equivalent to solving a system of first order partial differential equations with constant coefficients.
This is not possible without more restrictions on the $h_{ij}$. For instance, in the case $n=1$, you just have a single real-analytic function $h=h_{11}:\mathbb{C}\to(0,\infty)$ and want to find a holomorphic function $p=p_1$ such that $p'+\overline{p'}=h$. That is, you want $p$ such that $2\operatorname{Re}(p')=h$. This is possible only if $h$ is harmonic, since $p'$ will be holomorphic and the real part of a holomorphic function in one variable is harmonic. In fact, $h$ must be constant in this case, since a bounded-below harmonic function on all of $\mathbb{C}$ is constant.
I don't know how these restrictions generalize for $n>1$, though it seems a similar argument will show that $h_{ii}$ must be constant for each $i$.